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176-midterm-solutions-03-03-11 - Physics 176 Midterm Exam...

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Physics 176: Midterm Exam Solutions March, 2011 Most of the following answers are more detailed than what was necessary to get full credit. I hope the details will help you improve your problem solving skills and understand the physics better. Problems That Require Writing 1. Black holes are simple pure objects in that, no matter what kind of matter collapsed to form the black hole, only three numbers are needed to define its macrostate: the black hole’s mass M , its electrical charge Q , and its angular momentum L . For an electrically-neutral non-rotating black hole ( Q = 0, L = 0 ), the black hole’s entropy S depends only on M and is given by S = 8 π 2 kG hc M 2 (1) where k is Boltzmann’s constant, G is the gravitational constant, h is Planck’s constant, and c is the speed of light. (a) (5 points) Given that the energy of a black hole is its relativistic rest mass U = Mc 2 , derive a formula for the temperature T of a black hole in terms of its mass M . Does decreasing the mass of a black hole make it hotter or colder? Answer: The temperature can be deduced from the dependence of the entropy S ( U ) on the system’s energy U using the standard formula. We have: 1 T = ∂S ∂U (2) = ∂S ∂M × dM dU (3) = 8 π 2 kG hc · 2 M × 1 c 2 (4) = 16 π 2 kG hc 3 M, (5) so the answer is T = hc 3 16 π 2 GkM M - 1 . (6) Note how the entropy and temperature of a black hole involve fundamental constants from four different areas of physics: thermal physics ( k ), gravity ( G ), quantum mechanics ( h ), and special relativity ( c ). This makes black holes intriguing indeed. Since T M - 1 and mass is always a positive quantity (any material object resists acceleration), the temperature must also be positive for a black hole. Black holes are exotic objects but they can’t have a negative temperature like a paramagnet. Because T M - 1 , the smaller the mass of a black hole, the hotter it is. A sufficiently small black hole (about the mass of the Earth’s moon or less, 10 22 kg) will become hotter than 3 K the temperature of the cosmic blackbody radiation. We would then expect energy to transfer from the black hole to the surrounding photon gas if the black hole could transfer energy. This has been predicted to be possible by the physicist Stephen Hawking, who heuristically applied relativistic quantum mechanics to Einstein’s classical gravity theory to predict that all black holes emit light radiation from their event horizon (called “Hawking radiation”) and so can evaporate away into light if hotter than their surroundings. Further, the smaller the mass of a black hole, the faster it loses mass by Hawking radiation. The tiny small mass black holes predicted to occur 1
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in the Large Hadron Collider (based on the assumption that there are more than three spatial dimensions) are super hot and disappear within 10 - 10 s of their appearance, in a burst of gamma rays which would be the signature for their appearance. You can learn more about Eq. (6), called the Hawking radiation temperature, from the Wikipedia article with title “Hawking Radiation”.
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176-midterm-solutions-03-03-11 - Physics 176 Midterm Exam...

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